Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)
Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)
Solution and Explanation
Let A= \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)
We know that A = IA
so \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)= \(\begin{bmatrix}1&0\\0&1\end{bmatrix}A\)
\(\Rightarrow\begin{bmatrix}1&0\\1&1\end{bmatrix}=\begin{bmatrix}1&-1\\0&1\end{bmatrix}A\) (R1->R1-R2)
\(\Rightarrow\begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatrix}1&-1\\-1&2\end{bmatrix}A\) (R2->R2-R1)
\(\therefore\) A-1= \(\begin{bmatrix}1&-1\\-1&2\end{bmatrix}\)
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Concepts Used:
Invertible matrices
A matrix for which matrix inversion operation exists, given that it satisfies the requisite conditions is known as an invertible matrix. Any given square matrix A of order n × n is called invertible if and only if there exists, another n × n square matrix B such that, AB = BA = In, where In is an identity matrix of order n × n.
For example,
It can be observed that the determinant of the following matrices is non-zero.
